Assay for filtration of suspended particles in microporous membranes

ABSTRACT

A method of assessing a membrane, including calculating fluid dynamic characteristics of at least one of a membrane and a material to be passed through the membrane, where the material comprises particles; obtaining characteristic of at least one force acting on the particles of the material to be passed through the membrane due to the interaction between the particles and the membrane, the at least one force being an intermolecular force; combining the calculated fluid dynamic characteristic and the obtained characteristics to assess the flow of the material through the membrane; and optimizing at least one characteristics of the membrane in relation to the material. The membrane includes a plurality of rows and a plurality of teardrop structures arranged in the plurality of rows. The teardrop structures in each row are arranged at substantially the same angle with respect to an anticipated direction of flow through the membrane.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication No. 62/826,174, filed Mar. 29, 2019, which is incorporatedby reference as if disclosed herein in its entirety.

FIELD

The present technology relates to the field of porous polymer materials,and more particularly, to the performance of microporous membranes.

BACKGROUND

An important and unstudied aspect of porous materials is the linkbetween the microstructure and performance. Porous polymer materials areimportant in catalysis and in molecular separations such as syntheticmembrane filtration and chromatography. These materials arecharacterized by their microstructure such as pore size distribution(“PSD”), pore connectivity between pores, and surface reactivity. Porouspolymer materials are traditionally synthesized through empiricaloptimization of a phase inversion process. Pore formation occurs when apolymer solution undergoes a precipitation process involving temporaland local fluctuations in conditions that lead to a distribution ofmorphologies and the resultant PSD, in combination with surfacechemistry, controls the efficiency, selectivity, and capacity formembrane filtration and chromatography. In the current state-of-the-art,a desired PSD is targeted through qualitative correlations of parametersin a ternary phase diagram with the process parameters of a membraneproduction line. This approach is plagued by numerous deficiencies,including a lack of mechanistic understanding of pore formation duringthe phase inversion process, an absence of guidelines to select solventand non-solvent, and the impact of process specific variance on thefinal outcome. Such empirical optimization of porous polymer materialsis costly and time-consuming.

Since pressure-driven membrane processes such as reverse osmosis,ultrafiltration, and micro filtration are rate-limited processes, asopposed to equilibrium processes like distillation and adsorption,selectivity depends directly on the relative rates of transport fordifferent species through the membrane. Tracking the simultaneousindividual movement of these species inside a membrane to optimize themorphological structure, pore size distribution, and chemical nature isextremely challenging.

Besides microporous membranes, tracking particles inside porous media isof interest in depth filtration, chromatography, water treatment,secondary and tertiary oil recovery, and natural filtration ofmicroorganisms in subsurface aquifers. The aspect ratio of the mediadifferentiates all these 3-dimensional applications with microporousmembranes being essentially 2-dimensional thin films. Except forchromatography, all the other media are essentially inorganic and notsynthetic porous polymer materials, so the interactions between theparticles and the matrix are different. Extensive modeling offluid-particle transport in these 3D materials has been performed andincludes wall and particles interactions, such as long-range electricaldouble layer and van der Waals' forces. This is not the case withrespect to the measurement of particles experimentally within a porousmedium to understand how they interact with the medium and how theytravel within the medium. Also, most use well-defined geometric modelsfor the adsorptive media, like spheres, cylinders, and constrictedtubes. An exception is theoretical single and multiple particletrajectories in a 2-dimensional porous medium, the cross-section ofwhich was reconstructed from micro-CT scans of a real rock. The poresand particles were three orders of magnitude larger than those reportedhere, and the results were not compared with experimental measurements.Also, particles were forced to enter the medium in one of threeconduits, which differs from the present technology in which particlesare dragged by the fluid flow into any of the available conduits at theentrance of the porous medium.

Some have related the results of this multi-phase transport phenomena,by defining selectivity, to the concentrations of species in thepermeate relative to those in the feed. This lumped parameter approachis sometimes sufficient to characterize the global performance of aparticular membrane and process. However, there is a need for improvedapproaches to rationally determine how to improve the performance(selectivity and capacity) of a membrane through optimal design of themembrane structure and chemistry, since both selectivity and capacitydepend directly on these transport rates.

SUMMARY

Accordingly, a first embodiment of the present technology is directed toa method of assessing a membrane. The method includes the steps of:calculating fluid dynamic characteristics of at least one of a membraneand a material to be passed through the membrane, where the materialcomprises particles; obtaining characteristic of at least one forceacting on the particles of the material to be passed through themembrane due to the interaction between the particles and the membrane,the at least one force being an intermolecular force; combining thecalculated fluid dynamic characteristic and the obtained characteristicsto assess the flow of the material through the membrane; and optimizingat least one characteristics of the membrane in relation to thematerial.

In some embodiments, the step of calculating fluid dynamiccharacteristics includes computation of the fluid and particle dragmechanics associated with the material in at least two spatialdimension. In other embodiments, the step of calculating fluid dynamiccharacteristics includes computation of the fluid and particle dragmechanics associated with the material in three spatial dimension.

In some embodiments, the step of obtaining characteristics of at leastone force includes measuring the intermolecular forces between themembrane and the particles.

In some embodiments, the step of optimizing at least one characteristicsincludes optimizing the capture or release of particles by the membrane.

In some embodiments, the membrane includes a plurality of rows and aplurality of teardrop structures arranged in the plurality of rows.

In some embodiments, the teardrop structures in each row are arranged atsubstantially the same angle with respect to an anticipated direction offlow through the membrane.

In some embodiments, the membrane further includes that rows of theteardrop structures in which the structures are at an angle of 10°alternate with rows of the teardrop structure in which the structuresare at an angle of −10° relative to the anticipated direction of flowthrough the membrane. In some embodiments, the membrane further includesthat rows of the teardrop structures in which the structures are at anangle of 45° alternate with rows of the teardrop structure in which thestructures are at an angle of −45° relative to the anticipated directionof flow through the membrane. In other embodiments, the membrane furtherincludes that rows of the teardrop structures in which the structuresare at an angle of 70° alternate with rows of the teardrop structure inwhich the structures are at an angle of −70° relative to the anticipateddirection of flow through the membrane. In yet other embodiments, themembrane further includes that rows of the teardrop structures in whichthe structures are at an angle of 170° alternate with rows of theteardrop structure in which the structures are at an angle of −170°relative to the anticipated direction of flow through the membrane.

In some embodiments, the membrane is formed of a microporous hydrophilicpolymer material.

In some embodiments, the material to be passed through the membranecomprises a plurality of SiO₂ particles.

According to another embodiment of the present technology, a microporousmembrane is provided. The membrane includes a plurality of rows and aplurality of structures arranged in the plurality of rows, wherein thestructures in each row are arranged at substantially the same angle withrespect to an anticipated direction of flow through the membrane.

In some embodiments, the plurality of structures are teardropstructures. In some embodiments, the membrane is formed of a hydrophilicpolymer material.

In some embodiments, the membrane further includes that rows of thestructures in which the structures are at an angle of 10° alternate withrows of the structure in which the structures are at an angle of −10°relative to the anticipated direction of flow through the membrane. Insome embodiments, the membrane further includes that rows of thestructures in which the structures are at an angle of 45° alternate withrows of the structure in which the structures are at an angle of −45°relative to the anticipated direction of flow through the membrane. Inother embodiments, the membrane further includes that rows of thestructures in which the structures are at an angle of 70° alternate withrows of the structure in which the structures are at an angle of −70°relative to the anticipated direction of flow through the membrane. Inyet other embodiments, the membrane further includes that rows of thestructures in which the structures are at an angle of 170° alternatewith rows of the structure in which the structures are at an angle of−170° relative to the anticipated direction of flow through themembrane.

Further objects, features, and embodiments of the present technologywill be apparent from the drawing figures and below description.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is chart showing a diffuse double layer with Debye length as afunction of dilute ionic species concentration. The insert shows thepotential drop in solution from a wall potential, φ₀=37.7 mV.

FIG. 2 is a chart showing a voltage solution potential where the surface(key) is equal to log(V), and streamlines are for the velocity field.

FIG. 3 is a chart showing a laminar flow solution. The observed 5 (p.mmean pore diameter PES membrane segment has a thickness of 130 μm frominlet (y-axis, top) to outlet (y-axis, bottom) and width segment of 100μm (x-axis). FIG. 3A shows a 2D laminar flow field, and FIG. 3B shows apressure field, where pressure drops from a maximum of 50 Pa at the topface to nearly zero at the bottom face.

FIG. 4 is a chart showing an ATR-FTIR spectra of hot-pressed membranesat different temperatures. ATR-FTIR spectra of 0.2 μm mean pore size PESmembranes pressed at different temperatures. Control=Membrane asreceived (not pressed) at RT. Data were normalized using the peak at1,484 cm⁻¹ (aromatic C═C) as a reference.

FIG. 5 is an SEM image of PES microporous membranes. The left and rightcolumns are for 0.2 and 5 82 m mean pore size PES membranes,respectively. The top, middle, and bottom rows are face-on,cross-section, and face-on views, respectively.

FIG. 6 is a chart showing an ATR-FTIR spectra of different pore sizemembranes. ATR-FTIR spectra of 0.2 μm (bottom) and 5 82 m (top) meanpore size PES membranes. Data were normalized using the peak at 1,484cm⁻¹ (aromatic C═C) as a reference. The one peak difference at −1,640cm⁻¹ is likely due to the addition of an additive likeN-vinyl-2-pyrrolidone, often used in PES membranes, which has an intenseabsorbance at 1,665 cm⁻¹ (stretching vibration of C=0) or some residualsolvent like N,N-dimethyl acetamide, often used to dissolve PES, whichhas a characteristic peak at 1,638 cm⁻¹ (C=0 in the amide group).

FIG. 7 shows flow domain mapping. FIG. 7A is an SEM image of 5 82 mmicroporous PES membrane, side view. FIG. 7B is a serial block face(“SBF”) image with 100×130 μm mapped flow domain. FIG. 7C is acomputational liquid domain.

FIG. 8 is a schematic view of intermolecular forces. Particle-membraneinteractions were measured experimentally, while only particle-membraneinteractions were considered by the computational fluid and particledynamics analysis. Tracking particle intrusion into and attached ontothe internal pore structure of these PES membranes were obtainedexperimentally (SEM) and computationally.

FIG. 9 is a chart showing zeta potential measurement of PES membranes.Zeta potential estimates using both steaming potential (solid diamonds)and streaming current (open circles). The data (n=30) show that using ahot-pressed membrane, the two methods gave a similar result, independentof the channel height.

FIG. 10 is a chart showing surface interactions (adhesion) measured byAFM. FIG. 10A shows force maps between a SiO₂ particle and a SiO₂ waferas a function of trigger forces (1-2 pN) and KCl concentration (1-10-100mM). White pixels indicate high adhesion, while black pixels indicatelow adhesion (n=400). FIG. 10B shows histogram distributions of theadhesion forces of FIG. 10A for both data sets at 1 pN (black) and 2 pN(white) trigger force. Median values=54, 79, and 239 pN for 1, 10, and100 mM KCI concentrations, respectively. FIG. 10C shows force maps andhistogram distributions (insets) between a SiO₂ particle and ahot-pressed PES membrane, as a function of KCl concentration (0-1-10-100mM) (n=400). Median values=62, 138, 333, and 375 pN for 0, 1, 10, and100 mM KCl concentrations, respectively.

FIG. 11 is a chart showing repulsive (electrostatic) forces measured byAFM. Averaged (n=50) force-distance profiles measured by approaching aSiO₂ particle with a (FIG. 11A) SiO₂ wafer, and (FIG. 11B) hot-pressedFES membrane, as a function of KCl concentration (1-10-100 mM). Scanspeed=1 82 m/s; trigger force=1 nN.

FIG. 12 is a chart showing surface interactions measured by AFM.Schematic of (FIG. 12A) the adhesion and (FIG. 12D) the electrostaticrepulsive forces measured between a SiO₂ particle and (FIGS. 12B and12E, respectively) a SiO₂ wafer or (FIGS. 12C and 12F, respectively) ahot-pressed PES membrane, as a function of KCl concentration (1-10-100mM). FIGS. 12B-C show histogram distributions of the adhesion forces(n=400). FIG. 12B shows data sets at 1 nN and 2 nN trigger force. Medianvalues=56, 88, and 243 pN for 1, 10, and 100 mM KCl concentrations,respectively. FIG. 12C shows data set 1 nN trigger force. Medianvalues=138, 333, and 375 pN for 1, 10, and 100 mM KCl concentrations,respectively. FIGS. 12E-F show averaged force-distance profiles measuredby approaching a SiO₂ particle to the sample (n=50). Scan speed=1 82m/s; trigger force=1 nN. Data (circle) including error bars, and DLVOfits (black line). Fitting parameters were (FIG. 12E) [φ₀=−39.97 mV],[φ₀=−22.29 mV], and [φ₀=−9.33 mV], and (FIG. 12F) [φ₀=−59.40 mV,A=3.1.10⁻²¹ J], [φ₀=−28.15 mV, A ˜0 J], and [φ₀=−17.25 mV, A ˜0 J] for1, 10, and 100 mM, respectively.

FIG. 13 is a chart showing the flow and pressure fields inside acommercial 5 p.m mean pore size porous PES membrane. The observedmembrane segment has a thickness of 130 μm from inlet (y-axis, top) tooutlet (y-axis, bottom) and width segment of 100 μm (x-axis). FIG. 13Ashows a 2D laminar flow field, and FIG. 13B shows a pressure field,where pressure drops from a maximum of 14 kPa (˜2 psi) at the top faceto nearly zero at the bottom face. FIG. 13C shows average velocity (m/s)profiles across the 100 μm width at depths of 0, 40, 80, and 120 μm fromthe top face.

FIG. 14 is a chart showing local electric field in solution in theporous PES membrane of FIG. 13.

FIG. 15 is a chart showing the effect of van der Waals interactions onthe membrane of FIG. 13. 300 equally spaced 1 82 m diameter chargedparticles with a membrane wall potential of φ₀=−65 mV are released in a1 mM electrolyte solution, at the top surface where the driving pressureis set to 14 kPa (˜2 psi). Fluid mechanical forces include Stokes dragand Saffman lift, while particle-surface forces include electrostaticrepulsion (with DLVO double layer) (FIG. 15A) without van der Waal'sattraction or (FIG. 15B) with van der Waal's attraction to the surface.

FIG. 16 is a chart showing the flow, pressure, and electric fieldsinside a commercial 5 82 m mean pore size porous PES membrane. Theobserved membrane segment has a thickness of 130 μm from inlet (x-axis,left) to outlet (x-axis, right) and a width segment of 100 μm (y-axis).FIG. 16A shows 2D laminar flow field (below) and average velocity (m/s)profiles across the 100 μm width at depths of 10, 50, 90, and 130 μm(dashed lines) from the top face (left). FIG. 16B shows pressure field,where pressure drops from a maximum of 14 kPa (˜2 psi) at the top face(left) to nearly zero at the bottom face (right). FIG. 16C shows localelectric field in solution in the porous PES membrane.

FIG. 17 is a chart showing transient single particle tracking in themembrane of FIG. 13. 300 equally spaced 1 82 m diameter particles with amembrane wall potential of φ₀=−80 mV were released in a 1 mM electrolytesolution at the top surface where the driving pressure is set to 50 Pa.Fluid velocity is given by the color key on the right. Integratedparticle numbers (right histogram) at each horizontal position as afunction of depth along the membrane from inlet (top) to exit (bottom).2 82 m bins were used.

FIG. 18 is a chart showing the number of total, free, and adhered(stuck) particles with path length along the flow direction for theparticles tracked in FIG. 17. Integrated particle numbers as a functionof depth along the membrane from inlet to exit. 10 p.m bins were used.For the first bin at the top, 66.5% and 33.5% of 155 particles were freeand stuck, respectively.

FIG. 19 is a chart showing transient dual particle tracking in themembrane of FIG. 13. 150 1 μm diameter (blue) and 150 2 82 m diameter(red) particles were introduced together and simultaneously in a 1 mMelectrolyte solution in the area directly above the top membranesurface. The membrane wall potential was φ₀=−65 mV and the drivingpressure at the top face was 14 kPa (˜2 psi). Of the 300 releasedparticles: 235 particles were free (124 1 μm diameter particles and 1112 82 m diameter particles), 23 particles were adhered (or stuck) to theinternal surface (12 1 82 m diameter particles and 11 2 82 m diameterparticles), and 42 particles disappeared/permeated completely (14 1 82 mdiameter particles and 28 2 82 m diameter particles) after the run.Integrated particle fractions (of the total at that cross-section) as afunction of depth along the membrane from inlet to exit. 2 82 m binswere used.

FIG. 20 is a chart showing a binary system of particles in a membranewith three orientations of teardrops. FIG. 20A shows 10°, FIG. 20B shows70°, and FIG. 20C shows reversed 10° (=170°) teardrops rotated from thevertical.

FIG. 21 is a chart showing a binary system of particle tracking for thecommercial membrane and the computer-generated teardrop membrane. FIG.21A shows the commercial membrane, and FIG. 21B shows thecomputer-generated teardrop membrane with orientation rotated from theaxial direction of the flow of ±170°. 300 1 μm diameter and 150 2 82 mdiameter particles were simultaneously introduced in the region directlyabove the top membrane surface. The membrane wall potential was φ₀=−65mV, the number of elementary particle charges was z=100 andparticle-wall interactions included van der Waals forces (Hamakerconstant=6×10⁻²¹ J). The driving pressure at the top face was 14 kPa (˜2psi). Integrated particle fractions (of the total entering particles)are shown in the top and bottom bars (for particles of 1 and 2 82 mdiameter, respectively) as a function of depth along the membrane frominlet to exit. 5 μm bins were used. Black bars are for stuck particles.Particle rejections for FIG. 21A are R=0.53 (1 μm) and 0.49 (2 μm), andfor FIG. 21B are R=0.03 (1 μm) and 0.04 (2 μm). Rejection was calculatedfrom the particle fraction retained in the membrane at the end of thesimulation time relative to the total particles entering the porousmembrane. The images were taken after 4 ms. For the computer-generatedteardrop membrane, the simulation time was sufficient to complete thefiltration experiment. Hence, only “stuck” particle were present withinthe membrane and rejection was equivalent to the end value in a depthfiltration experiment. For the commercial membrane, however, both stuckand slow moving free particles remained in the membrane at the selectedtime for calculating rejection.

FIG. 22 is a chart showing a binary system of particle tracking incomputer-generated teardrop membranes. Membranes: Computer generated,staggered array of teardrop obstructions with three orientations rotatedfrom the axial direction of the flow: (FIG. 22A)±170° (FIG. 22B)±10° and(FIG. 22C)±45°. 300 1 82 m diameter and 150 2 82 m diameter particleswere simultaneously introduced in the region directly above the topmembrane surface. The membrane wall potential was φ₀=−65 mV, the numberof elementary particle charges was z=100 and particle-membraneinteractions included van der Waals forces (Hamaker constant=6×10⁻²¹ J).The driving pressure at the top face was 14 kPa (˜2 psi). Integratedparticle fractions (of the total entering particles) are shown in topand bottom bars (for particles of 1 and 2 82 m diameter, respectively)as a function of depth along the membranes from inlet to exit for thethree embodiments. 5 um bins were used. Particles located near theentrance (left) for all three runs are stuck (statistics unavailable).The images were taken after 1.02, 0.86, and 0.94 ms, respectively, justbefore the first particle permeated through the membrane. The simulationtime was 4 ms.

FIG. 23 is a chart showing parameter analysis for varying the Hamakerconstant of the membrane/particle interaction. These data were measuredfor flow in 5 computer-generated teardrop membranes with orientationrotated from the axial direction of the flow of ±170°. The Hamakerconstant for the particle-membrane van der Waals interaction wasincreased from 6×10⁻²¹ to 6×10⁻¹⁷ J. 300 1 82 m diameter and 150 2 82 mdiameter particles were simultaneously introduced in the region directlyabove the top of each membrane surface. The membrane wall potential wasφ₀=−65 mV, the number of elementary particle charges was z=100. Thesimulation time was 4 ms. FIG. 23A shows particle rejection for both 182 m diameter (small circle) and 2 82 m diameter (large circle)particles. FIG. 23B shows average axial particle velocity. FIG. 23Cshows particle residence time for the slowest (diamond) and fastest(square) particle to reach the exit.

FIG. 24 is a chart showing the comparison between the commercial andhypothetical teardrop structure membranes. Membrane: Computer generated,staggered array of teardrop obstructions. Individual rows of teardropswere rotated ±170° from the vertical. 300 1 82 m diameter and 150 2 82 mdiameter particles were introduced together and simultaneously in a 1 mMelectrolyte solution in the area directly above the top membranesurface. The membrane wall potential φ₀=−65 mV, the particle charge wasZ=−100 and particle-particle interactions included van der Waals(Hamaker constant, A=10⁻¹⁵ J). The driving pressure at the top face was14 kPa (˜2 psi) and the maximum velocity was 0.2 m/s.

FIG. 25 is a chart showing a qualitative comparison between experimentsand model predictions. FIG. 25A shows an overlay of SEM image with EDSsignal collected for silicon, shown at sufficient magnification toidentify individual 1 82 m SiO₂ particles, within the cross section ofthe PES membrane, distinguished from the membrane nodules. FIG. 25B isan SEM image of the entire membrane thickness. FIG. 25C shows EDS signalcollected for silicon across the entire membrane. FIG. 25D is anintegration (orthogonal to flow) of collected EDS signal in FIG. 25C,reflecting the concentration of silica particles as normalized by thelargest peak (normalizing factor=417 counts). Data is binned every 2 82m (40 pixels) along the flow axis. FIG. 25E is a computer-generatedprofile of predicted particle concentration as a function of membranedepth (y-coordinate) using computational fluid mechanics normalized bythe highest particle count (normalizing factor=67 particles). In FIGS.25D-E, the data were normalized by the highest peak in the data set.

DETAILED DESCRIPTION

Accordingly, embodiments of the present technology address the importantlink between the microstructure of a membrane and its filtrationperformance. In some embodiments, a “reverse process” is used, in whichthe membrane performance is first simulated and optimized in silico by acomputational fluid dynamics tool and then a preferredcomputer-generated structure is synthesized. 2D computational fluid andparticle drag mechanics are combined with particle and membrane forcemeasurements in aqueous solutions containing inorganic ions to studyparticle intrusion and capture in microporous commercial polymer andcomputer-generated teardrop membranes. Fits of the DLVO theory toforce-distance profiles obtained membrane surface potentials needed forthe computations. In silico predictions of particle intrusion for acommercial membrane qualitatively agree with experimental filtrationmeasurements using scanning electron microscopy with particle trackingvia energy dispersive X-ray spectroscopy. Highlighting the poor flowfield uncovered several dominant inhomogeneous 2D flow conduits withlarge unused regions of the internal pore structure. To guide improveddesign, new computer-generated microporous teardrop structures that canequalize the flow field, adjust the tortuosity of the flow path, andvary the reactivity of the surface were tested in silico. The mainassumptions of the computational model were that 2D flows are a validdescription of 3D flows, all forces were applied at the particle centerof mass, and forces were calculated based on the physical diameter ofthe spherical particles. Relatively large pores (˜5 micron) and largeparticles (˜1 micron) were selected for easy detection and analysis.Preferably, the computational fluid and particle flow analysis and theinter-surface forces scale independently with size and applies at allclassical dimensions (i.e. for nano, ultra, and microfiltration).Assumptions for the intermolecular force measurements were thatelectrostatic and van der Waal's forces dominated and hence that theDLVO theory was valid, and that the zeta potential values were close tothose at the wall (i.e. surface potential). In particular, the DLVO wasapplied to ideal geometries: a sphere (i.e. AFM probe) near to a flatsurface (i.e. either a silica wafer or a hot pressed PES membrane). Thiscomputational fluid mechanics-based tool can be used to characterizemembranes for separation performance and guide improved design,synthesis and testing of new microporous membranes.

Accordingly, the difficulty of designing a synthetic polymer membranewith a desirable pore size distribution, specified morphology, andsurface chemistry is a major deficiency that seriously limits progressin optimizing filtration selectivity and capacity (permeation flux). Tohelp address this challenge, the concept of selectivity and hencerelative transport rates of competing solutes (or particles) into andthrough a membrane needs quantitative analysis. In some embodiments, 2Dcomputational fluid and particle drag mechanics are combined withintermolecular force measurements to study particle intrusion andattachment in microporous polymer membrane (i.e. microfiltration) pores.In other embodiments, 3D computational fluid and particle drag mechanicsare combined with intermolecular force measurements to study particleintrusion and attachment in microporous polymer membrane (i.e.microfiltration) pores. In some embodiments, the predictions from thistheoretical approach are combined qualitatively with experimentalmeasurements of particle intrusion into microporous polymer membranesusing scanning electron microscopy with particle tracking via energydispersive X-ray spectroscopy.

Some embodiments of the present technology provide new insight oninternal particle capture with qualitative agreement with theexperiments, and the existence of several dominant 2D flow conduitsinstead of even fluid flow with large regions of the internal porestructure unused. Some embodiments of the technology show 2D resultsshowing that the internal morphological structure of commercialmicroporous membranes could be poorly designed for optimal particlecapture or release and hence selectivity and permeation flux. Someembodiments of the technology include improved filtration performance,via one or more synthetic morphological structures based on a teardropdesign that show even exit axial flow across the horizontal axis andseparation between small and large particles, both of which were notobserved for simulated transport in commercial microporous membranes.

According to some embodiments of the present technology, the drag andadhesion of point-particles in a complex 2-dimensional flow-field aretracked through a realistic pore structure of a microporous commercialmembrane using wall potential from zeta potential measurements. Theinteraction forces are calculated as if the particles had physical size,charge, and mass. However, only the center of mass of the particles aretracked, not their surface. These interactions are obtained, in someembodiments, using zeta potential and atomic force microscopy in forcemode measurements as a function of ionic strength. In some embodiments,the particle hold-up data predicted by the computational model iscompared with those of silica particle intrusion measurements.

In some embodiments of the present technology, a finite element modelwas developed to track the paths of particles through two commercialpoly(ether sulfone) (“PES”) membranes (with 0.2 and a 5 82 m mean poresize) and to simulate the interactions of the particles with one anotherand with the membrane surface. For the commercial PES membranes, thedomain geometry was derived from their SEM micrographs. Additionally,hypothetical membrane structures were developed with teardrop designs.AutoCAD software was used to trace images from SEM images and separatethe solid membrane regions from the open pore space. The pore space wasthen filled with water containing a 1:1 binary salt, like KCl, inconcentrations ranging from 0.1-100 mM. The model coupled hydrodynamicswith electrostatics (−65 mV (from zeta potential measurements) andestimated −80 mV wall potential), van der Waals interactions, and dilutespecies transport to describe the flow field, the electric field and thedistribution of ions in the domain, as shown in FIGS. 1-2. Particleswith point charge (−100 mV) were periodically introduced into the flowfield outside the top face of the membrane and subjected tohydrodynamic, electrostatic, and van der Waals forces.

In some embodiments, the flow field was simulated as pressure drivenflow with a very low pressure drop of 50 Pa, as shown in FIG. 3, and arealistic pressure drop of 14 kPa (˜2 psi) derived from the membranefiltration experiments. A slip condition was used on the overall domainsidewalls, since the SEM image represented only a small slice of thetotal membrane. The electric field within the pore space of the membranewas calculated by specifying zero charge boundary conditions on theoverall, external boundaries of the domain and using the experimental,zeta potential measurements reported herein to set the potential on allthe internal membrane surfaces. The ion distribution was simulated usingzero flux boundary conditions on all surfaces and setting an initialsalt concentration in the water with a constant salt concentration onthe upper boundary leading into the top membrane surface. A steady-statesimulation was first performed to set the underlying flow, electric, andconcentration fields prior to introducing the particles.

In some embodiments, the simulations were run in 2D to develop the modelformulation in a simpler geometry first, understand what forces wereimportant in the system, determine how big the models would get whenusing a real membrane geometry, determine the minimum feature scalesneeded to include from the membrane geometry, and determine how muchtime it would take to simulate the interactions of many individualparticles with the membrane in detail. In some embodiments, the flowfield is solved for in a 3D geometry by deriving 3D geometries from theSEM slices, importing them into the simulation, and creating a 3Dsimulation of the entire system with a volume roughly 100 microns on aside.

In some embodiments, the microporous membranes used were hydrophilic PES0.2 and 5 82 m mean pore size membrane. The monodispersed silica (SiO₂)particles were 5% w/w in water with a diameter of 0.25 and 1 82 m. ASiO₂ wafer was used for the force measurements. Ultra-pure water(resistivity ˜18 MS2) was used for all the experimental work.

Scanning electron microscopy (“SEM”) was used to analyze the morphologyof the top and bottom faces and the cross-section (edge) of thecommercial microporous structure of the PES membranes. In someembodiments, to facilitate imaging of the membrane and reducebeam-induced damage, a beam accelerating voltage of 5 kV was used andthe non-conductive membranes were sputter coated using an Au/Pd alloy.The SEM is capable of recording images with up to 4000 pixels in thevertical direction, meaning that the images typically contain nearly 30pixels per micron across the entire membrane (130 μm thickness), andthus the particles were identifiable from images of the entire membrane.However, the membrane also has 1 82 m in diameter round nodules, whichmade more complex discriminating between particles and membrane.Unfortunately, the membrane and the particles are also similar in atomicweight, suggesting that discrimination through background electrons(“BSE”) was also not practical, with most of the BSE signal variationcoming from sample geometry rather than atomic number. However, theparticles are made of SiO₂, and while the membrane contains oxygen, itdoes not contain silicon. Thus, the SiO₂ particles were discriminatedfrom the PES nodules using characteristic X-rays using Energy DispersiveX-ray Spectroscopy (“EDS”).

In some embodiments, EDS maps were generated to distinguish thepositions of SiO₂ particles from membrane nodules, since the siliconecharacteristic X-ray peak does not overlap significantly with the X-raypeaks of the PES membrane of the Au/Pd coating. The EDS maps wereproduced with the entire thickness in view (130 μm thickness) andcontained around 2,000 pixels in the axial (flow) direction, whichcorresponded to about 15 pixels per micron across the entirety of themembrane. As the beam was scanned across the sample, individual spectrawere generated for each pixel. The spectra consist of characteristicX-ray peaks overlaid on top of the background X-ray radiation (calledbraking radiation). The software then provides a pixel by pixeldetermination of the presence of Si atoms by comparing the X-raybackground to the Si peak. Stochastic variations in the background X-rayintensity can lead to false positives, but the result is sufficient fordistinguishing particles from the PBS membrane. Since the penetrationdepth of the particles into the membrane was of interest, the Sielemental maps were integrated orthogonally to the flow direction of themembrane. Each point in the integration provides an estimate of the areaof the membrane covered by particles for each line of the Si elementalmap along the direction orthogonal to the flow direction. Theseintegrations were then binned, reducing the resolution along the flowdirection, but also reducing the noise in the data.

In some embodiments, an SEM equipped with a SBF/SEM set-up was used forsequential imaging of the membranes in cross section. An in situultramicrotome inside the SEM chamber and a solid-state directionalbackscatter detector attached to the pole piece allow for the sequentialsectioning and imaging of the resin embedded membrane block-face. Toincrease the backscatter signals, in some embodiments, membranes werestained with osmium tetroxide for 2 hours. They were then embedded in anepoxy resin and cured in oven at 70° C. overnight. The samples embeddedin resin were fixed to an aluminum stub with epoxy glue. The block wastrimmed with an ultramicrotome into a cubic shape. To reduce chargingduring imaging, the lateral sides of the block face were coated withcolloidal silver glue and the block face was sputter-coated with 10 nmplatinum/palladium. In some embodiments, the serial block face imageswere acquired in an automated mode and in low vacuum mode under thefollowing conditions: accelerating voltage 3 kV, beam current 100 pAmp,resolution 1,012×884, pixel size 195 nm, slice thickness 100 nm, andchamber pressure 40 Pa.

In some embodiments, force measurements were performed using an atomicforce microscope, and the collected data were analyzed. The membranesand silica wafers were scanned in force mapping mode using siliconnitride cantilevers carrying a 1 82 m silica sphere, and nominal springconstants of 20 or 60 pN/nm. The cantilevers were calibrated before eachexperiment. A force map data set consisted of an array of 400 (20×20)force measurements, scanning in contact mode an area of 20×20 μm², witheach pixel point spanning an approximate width of 1 82 m in both X and Ydirections. Different parameters were varied: (i) the trigger force was0.2-0.5-1 nN; (ii) the scanning speed was 0.5-1-2 82 m/s; (iii) thesamples were immersed in H₂O or 1-10-100 mM KCl in H₂O. All measurementswere performed at 22° C.

In some embodiments, membrane surface zeta potential was determined fromthe measured streaming potential and/or streaming current using acommercial electrokinetic analyzer. Two 20×10 mm² hot pressed or asreceived membrane samples were fixed on the rectangular planar sampleholders of an adjustable gap cell using double-sided adhesive tape.Before each measurement, the samples were rinsed 5 times with theworking electrolyte solution. Four measurements were collected for eachoperating condition. All measurements were performed at 22° C., the gapdistance was kept constant and equal to ˜100 μm unless specified, theelectrolyte solution was 1 mM KCl in H₂O, pH ˜6.5.

In some embodiments, each PES membrane sample was cut into 2×2 cm²squares and sandwiched between two kapton films. Stainless steel platesand Carver Press were equilibrated at the desired temperature for atleast 15 minutes. Then, each membrane was pressed under 2 ton load for 5minutes at 150° C., 200° C., and 250° C., as shown in FIG. 4.

In some embodiments, Fourier Transform Infrared Spectroscopy(“ATR-FTIR”) was used to detect the structure of PES membranes. FTIRspectra were obtained at a 0.48 cm⁻¹ data spacing, and 16 scans wereperformed per sample in the wavenumber range of 400-4,000 cm⁻¹. Aquadratic function was fitted to each spectrum and subtracted to performbaseline correction.

In some embodiments, filtration experiments with 1 82 m silica particleswere performed using a stirred cell. Membrane coupons were cut to fitthe cells. A 10 mL particle solution was prepared by diluting the 5% w/wstock particle solution in the working buffer (e.g., H₂O or 1-10-100 mMKCl in H₂O) 1,000-fold and sonicated 2 minutes in a water bathsonicator, to disperse the particles. Gentle magnetic stirring wasincluded to minimize concentration polarization. Filtration wasperformed at 14 kPa (˜2-3 psi), using pressurized N₂. For mixtures, 0.25μm silica particles were added to the suspension of 1 82 m silicaparticles for the same total concentration of particles as with just thesingle particle runs.

Experiments to characterize and define the pore morphology and toestimate the intermolecular forces with changing salt (KCl)concentration, such as the surface energy (i) of PES membranes, (ii)between SiO₂ particles, and (iii) between SiO₂ particles and PESmembranes, were needed for the computational predictions of fluid andparticle movement passing through a microporous PES membrane. In someembodiments, computational fluid and particle dynamics with double layereffects (DLVO theory) to track particle intrusion into and attached ontothe internal pore structure of these PES membranes, were used toestimate mean particle number as a function of distance from the topmembrane surface and dominant flow paths within the membrane. Aqualitative comparison between the computational predictions and theexperimental results of the mean particle number obtained bymicrofiltration of a SiO₂ particle suspension was performed.

Top face-on, cross-section (edge), and bottom face-on SEM images of the0.2 and 5 82 m mean pore size PES membranes are shown in FIG. 5.Although the membranes are similar in chemistry, as shown in FIG. 6,their morphologies are different. The morphology of the μm mean poresize membrane appears to be similar for top, edge and bottom views,while for the 5 82 m mean pore size membrane these views show adifferent morphology. A perforated skin is present on the top and thebottom face, while a nodular porous structure is observed for thecross-section for the 5 82 m mean pore size membrane. A 2D cross sectionof each PES membrane was then used as an image template for thecomputational laminar fluid flow and particle drag, as shown in FIG. 7.

FIG. 8 is a schematic showing the intermolecular forces involved in someembodiments of the analysis. Regarding the surface energy of the 5.0 μmmean pore size PES membrane: zeta potential measurements of microporousmembranes using an electro-chemical test cell gave questionable resultswith the 5 82 m mean pore size microporous membrane. In particular, whencomparing zeta potentials obtained from streaming potential versusstreaming current, the measurements were inconsistent between the twomethods due to undesired convective flow in the porous membranes. As aresult, these measurements were re-run at 1 mM KCl as function of thegap-width with a non-porous hot-pressed PES membrane and found that thetwo methods (streaming potential and streaming current) gave similarconsistent results, as shown in FIG. 9. The streaming potential andstreaming current gave zeta potential values of −64.5±5 and −65.5±4.3mV, respectively. Thus, for the computation of particle-surfaceinteraction, −65 mV and −80 mV were selected as estimates of themembrane surface potential of the internal pore surface.

In some embodiments, the surface energy between SiO₂ particles wasobtained with AFM-FM between a 1 82 m mean diameter SiO₂ sphere and aSiO₂ wafer as a function of trigger force and increasing KClconcentrations. As seen in FIGS. 10A-B, the trigger force had littleeffect on the force measurements. However, the adhesion forces (i.e.jump-out force on pull-out) increased with increasing saltconcentration, as shown in FIG. 10B. The repulsive force as the surfacesapproached between a SiO₂ sphere and a SiO₂ wafer as a function ofincreasing KCl concentrations is summarized in FIG. 11A. The amplitudeand the extent of the repulsive forces increased with increasing KClconcentration. In calculating particle-particle interactions (andpossible aggregation) during intrusion, these measurements are neededwhen point size particles are replaced by particles with specifieddiameters. Here, the point particles are given a charge but do not asyet account for particle-particle steric interactions.

In some embodiments, the surface energy between SiO₂ particle and a PESfilm was obtained with AFM-FM between a 1 82 m mean diameter SiO₂ sphereand a PES sheet as a function of increasing KCl concentrations. As seenin FIG. 10C, the adhesion forces increased substantially with increasingsalt concentration. The repulsive force as the surfaces approachedbetween a SiO₂ sphere and a PES hot-compressed film as a function ofincreasing KCl concentrations is summarized in FIG. 11B. In someembodiments, a hot-compressed PES sheet was used to reduce datavariability caused by the membrane roughness. Once again, the amplitudeand the extent of the repulsive forces decreased with increasing KClconcentration.

In some embodiments, the median adhesion forces between the SiO₂particle and a PES film at 100 mM KCl were 136 pN greater than thatbetween the SiO₂ particle and a SiO₂ wafer, suggesting the adhesion tothe PES surface was more intense than particle-particle interactions.Also, the repulsive forces and extent of forces into the fluid werehigher and further, respectively, between the SiO₂ particle and a PEShot-compressed film as compared with that between the SiO₂ particle anda SiO₂ wafer, again suggesting the repulsion from the PES surface wasmore intense than between particles. Thus, once particles adhered to thePES (internal) surface, overcoming their adhesive force to dislodgethese particles requires more energy than detaching two SiO₂ particlesfrom each other.

In some embodiments, the surface forces and energies betweenparticle/particle and particle/PES film were measured in aqueoussolutions using AFM-FM, with a 1 82 m diameter SiO₂ particle attached toa cantilever and a SiO₂ wafer (substitute for SiO₂ particle) or ahot-compressed PES membrane (to obviate pores and roughness),respectively, as shown in FIG. 12.

In some embodiments, the adhesion forces, i.e. jump-out force onpull-out, increased with increasing KCl concentration for both SiO₂particle/SiO₂ wafer and SiO₂ particle/PES membrane, while it wasindependent of trigger force, as shown in FIGS. 12A-C. The medianadhesion forces between SiO₂ particle/PES membrane at 100 mM KCl were˜136 pN greater than that between SiO₂ particle/SiO₂ wafer. Thus, onceparticles adhered to the PES (internal) surface, overcoming theiradhesive force to dislodge these particles requires more energy thandetaching two SiO₂ particles from each other.

In some embodiments, the amplitude and extent of the repulsive force(i.e. positive curvature during the cantilever approach to the sample)decreased with increasing KCl concentration, indicative of electrostaticinteractions, for both SiO₂ particle/SiO₂ wafer and SiO₂ particle/PESmembrane, as shown in FIGS. 12D-F. Hence, the DLVO theory for a spherenear a flat surface fitted well to the force/SiO₂ particleradius-distance curves. In particular, for the SiO₂ particle/SiO₂ waferembodiment shown in FIG. 12E, the results showed that the force/particleradius-distance curves were well described by considering only theelectrostatic repulsion between the two negative SiO₂ surfaces andneglecting the van der Waals contribution. This shows the lack ofadhesion at small separation between silica surfaces in aqueouselectrolyte solutions: the hydroxylation of silica surfaces exposed towater for an extended time results in the formation of silicic acidgroup, carrying a negative charge. While SiO₂ particle/particleinteraction was neglected in some embodiments, the estimated surfacepotential was used to calculate the number of elementary charges on theSiO₂ particle surface: ˜132,000, 100,000, and 62,000 for 100, 10, and 1mM, respectively. The 1 82 m diameter SiO₂ particles used in someembodiments are large (on a molecular scale) and their surface chargesets up an electric field about the particle that is similar to thefield established by the membrane surface. In some embodiments, theelectrical force driving the particles toward or away from the membranesurface is proportional to the field strength in the liquid medium andthe charge on the particle. Given that the Debye lengths are so small,significant electrostatic forces only exist within a region that isextremely close to the membrane surface (i.e. a few 10s of nanometers).Once a particle gets in close proximity to the membrane, intermolecularforces begin to dominate over hydrodynamic forces, allowing theparticles to stick to the surface. For the SiO₂ particle/PES membraneembodiment shown in FIG. 12F, at KCl=1 mM the fitting parameters of theDLVO theory were: surface potential, φ₀=−59.9 mV and Hamaker constant,A=3.1×10⁻²¹ J.

The surface potential obtained by the DLVO theory was in good agreementwith independent zeta potential measurements using an electro-chemicaltest cell. Two methods, streaming potential and streaming current, gavecomparable zeta potential values of −64.5±5 and −65.5±4.3 mV,respectively, at KCl=1 mM. Thus, in preferred embodiments, for thecomputation of SiO2 particle/PES surface interactions, −65 mV and −80 mVare selected as estimates of the membrane surface potential of theinternal pore surface.

Modeling transport processes involved in membrane filtration is acomplex problem occurring over many length and time scales. In someembodiments, the aim is to simulate particle transport and hold-up in amicroporous PES membrane from the pore-level (0.2 and 5 82 m mean poresize) to the full membrane thickness (130 μm). Throughout, the lengthscales of ionic solutions are well within the continuum domain and aremodeled as continuous fluids. Water with an equimolar concentration ofcations and anions is used to simulate a DI water-KCl salt solution.

SiO₂ particles on the order of 0.25 and 1 82 m approach 5-20% of themean pore diameter, which challenges the continuous fluid approximation.However, in these embodiments, particles were points without size, andthis approximation was met. Newtonian forces act on the imaginaryparticles causing their acceleration. Stokes's drag arising from thefluid-particle interaction is the primary particle driving force. Liftforces, including wall-induced lift, act to perturb particles from fluidstreamlines. Close to the pore walls, forces arising from the surfaceelectrostatic potential are simulated according to the Debye length fordilute electrolyte solutions. The discreet particles carrying a chargeCoulomb and van der Waals forces are simulated to account forinter-particle forces.

In some embodiments, the methodology for simulating this type offiltration process is by coupling together several numerical solutionsin a piecemeal fashion. A static solution to Stokes's flow through aconformally mapped membrane geometry is the core of the simulation andgiven by:

0=Δ·{−ρ

+μ[∇

+(∇

)^(T)]}+

  (1)

0=ρ∇·(

)   (2)

In some embodiments, the flow domain geometry is directly modeled from2D SEM images, as shown in FIG. 7. The solution for the velocitydistribution and applied pressure for the laminar flow sub-model for the0.2 μm mean pore diameter PES membrane is shown in FIG. 3. Couple tothis solution are several sub-models. In some embodiments, a chemicalspecies transport simulation model is used to simulate a weak 1:1electrolyte solution. The ionic species from this simulation are coupledto an electrostatics simulation model to determine the penetration depthof the PES surface charge into the liquid domain. The resultingsub-model allowed for the Debye length from the pore walls to be afunction of ionic concentration as follows:

$\begin{matrix}{\frac{1}{\lambda} = {\sqrt{\frac{2e^{2}p_{\infty}^{c}}{{\epsilon\epsilon}_{o}{kT}}} = \sqrt{\frac{4e^{2}C_{0}}{{\epsilon\epsilon}_{o}{kT}}}}} & (3)\end{matrix}$

The parametric simulation shown in FIG. 5 was verified against thebenchmark value of 0.31 nm at a molarity of 1. Solution voltage resultson applying the simulation strategy to the 0.2 p.m mean pore diameterPES membrane geometry are shown in FIG. 8.

In some embodiments, fluid and particles drag through a commercialmicroporous membrane was analyzed. In some embodiments, the flow fieldas a function of pressure was determined. From a SEM image of the 5 82 mmean pore size porous membrane, an area with a thickness of 130 μm frominlet to outlet (y-axis) and a width segment of 100 μm (x-axis) wasselected. FIG. 13A represents a 2D laminar flow field with flow from topto bottom through a 2D porous structure of a 5 82 m mean pore diameterPES membrane obtained from SEM showing uneven flow through a few majorconduits. The pressure-drop from a maximum at the top face of the 5 82 mmean pore diameter PES membrane of 14 kPa (˜2 psi) is presented in FIG.13B, where 50% drop occurred in the bottom 30% of the flow path. Thisnonlinear pressure drop is likely due to the presence of the perforatedskin at the bottom face. The maximum Reynolds number (“Re”) for the flowwas approximately:

$\begin{matrix}{{Re}_{d,\max} = {\frac{vD}{v} = {\frac{0.9{\frac{m}{s} \cdot 5}\mspace{14mu}\mu\; m}{9 \times 10^{- 7}\frac{m^{2}}{s}} = 5}}} & (4)\end{matrix}$

Thus, the Re number varied from 0-5 within the membrane. A similaranalysis at very low pressure drop of 50 Pa gave the similar images tothose in FIGS. 13A-B with flow velocities of the order of ˜3 mm/s. InFIG. 13C, the axial velocities horizontally across the image at 0, 40,80, and 120 μm along the flow path from top to bottom are shown. Onlythree major flow conduits are observed suggesting very poor flowdistribution. These 2D highways of flow suggest that optimization of theinternal morphology for flow is needed.

The relative voltage potential (log (φ/φ₀), where (φ₀=−65 mV is thesurface potential obtained from zeta potential measurements) in solutionin the same membrane cross-section as that shown in FIG. 13, ispresented in FIG. 14. The electrical potential in some solution regionsnear the top 30% of membrane thickness approaches very low values (˜0mV), while in others near the bottom face, it is close to φ₀, i.e. log(φ/φ₀˜1.0)→0. For the former regions, the electrical potential insolution can be neglected, while for the latter regions it should beincluded when the particles are very close to the pore wall (<100 nm).Farther away (>100 nm), the potential field will not perturb particles.

In some embodiments, charged particle drag in the absence and presenceof van der Waals attraction to the surface was determined. 300 randomlyspaced (over an area of 125×5 μm²) 1 82 m diameter particles with apoint potential of φ₀=−100 mV at an electrolyte concentration of 1 mMwere released at the top surface of the same membrane cross-section asused in FIGS. 13-14 under a transmembrane pressure of 14 kPa (˜2 psi) inthe absence and presence of van der Waals attraction to the surface, asshown in FIGS. 15A-B. Their horizontal positions were randomly selected,and they were given initial velocities and trajectories dictated by thelocal flow field. The particles were then subjected to drag, electrical,and van der Waals forces. As shown in FIG. 15B, in the presence of vander Waals attraction, far less particles intruded into the microporousmembrane, but they were rather trapped at the entrance to the membraneat the top surface.

In some embodiments, from a SEM image of the 5 82 m mean-pore-sizeporous membrane, an area with a thickness of 130 μm from inlet to outlet(x-axis) and a width segment of 100 μm (y-axis). FIG. 16A shows the 2Dlaminar flow field through the porous structure. The left-hand side ofthe figure represents the top face of the membrane with inlet flow. Thefigure highlights the uneven flow distribution dominated by a few majorconduits that condense into one dominant fast velocity near the backface. The maximum pore Reynolds number (Re_(pore)) for the flow wasapproximately:

$\begin{matrix}{{Re}_{{pore},\max} = {\frac{v_{{ma}x}D_{pore}}{\upsilon} = {\frac{0.9{\frac{m}{s} \cdot 5}\mspace{14mu}\mu\; m}{9 \times 10^{- 7}\frac{m^{2}}{s}} = 5}}} & (5)\end{matrix}$

In some embodiments, the average Re_(pore) was ˜20 fold smaller, basedon the average velocity of 0.04 m/s. Only three major flow conduits areobserved. These represent the paths of least resistance through themembrane and correspond to regions having the least amount of PESmembrane material. These 2D flow highways suggest that furtheroptimization of the internal morphology for flow is required to enhancethe capacity of the membrane.

The pressure profile through the membrane is presented in FIG. 16B. Thedrop is overall linear except for the regions within about 10 μm of theentrance and exit of the membrane. In those regions, flow restrictionsdue to surface skin layers increase the local pressure drop. Thisbehavior correlates well with the number in inlet and outlet flowchannels observed in FIG. 16A.

The relative voltage potential (log φ/φ₀), where φ₀=−65 mV is thesurface potential obtained from zeta potential measurements) in solutionis presented in FIG. 16C. The electrical potential drops offexponentially moving away from the membrane wall. In most regions, theelectrical potential in solution can be neglected (˜0 mV) everywhereexcept within ˜100 nm of the statically charged wall.

In some embodiments, transient single particle tracking was performed.300 randomly spaced 1 82 m diameter particles with a wall potential ofφ₀=−80 mV at an electrolyte concentration of 1 mM were released alongthe top surface every ⅓ s for a total of 1 s (i.e. total of 900particles released). The same membrane cross-section as that used inFIGS. 13-15 was selected for particle tracking, as shown in FIG. 17.Since the particles hardly entered the pores in FIG. 15, the appliedpressure was reduced substantially to 50 Pa and the membrane wallpotential was decreased to φ₀=−80 mV to reduce wall adhesion. Onceagain, just as seen for the 14 kPa transmembrane pressure in FIGS.13-15, several dominant high velocity flow conduits appear. Manyparticles enter the membrane pores and are dragged along in these highvelocity flow conduits. Integrated particle numbers as a function ofdepth along the membrane flow path from inlet to exit is shown in FIG.17 (2 μm bins were used). Some particles were held-up in slow- orno-flow dead-ended pores. Here the hydrodynamic forces are ofteninsufficient to dislodge particles held on the walls or that collect inthe small pores. The number of total, free, and adhered (stuck)particles with path length along the flow direction is summarized inFIG. 18 (10 μm bins were used). For the first 10 μm at the top of themembrane, of 155 particles within this membrane porous region, 66.5% and33.5% were free and stuck, respectively. Of the total 900 (100%)particles entering the membrane, 493 (55%), 236 (26%), and 171 (19%)were free, stuck, and disappeared from view, respectively. These resultssuggest that particle capture was mostly near the top of the membrane.

In some embodiments, transient dual particle tracking was performed. 1501 p.m diameter and 150 2 82 m diameter were introduced together andsimultaneously in the area directly above the top membrane surface. Amembrane wall potential of φ₀=−65 mV and a transmembrane pressure of 14kPa (˜2 psi) with van der Waals attraction to the surface was selected.Also, all particles had the same repulsive negative charge, inqualitative agreement with AFM force measurements in FIG. 10B. Theirhorizontal positions were randomly selected, and they were given initialvelocities and trajectories dictated by the local flow field. Theparticles were then subjected to drag, electrical, and van der Waalsforces. When the particles approached the membrane to the point whereelectrostatic and van der Waals forces overcame the hydrodynamic forces,they stuck to the surface.

The model kept track of each particle and whether that particle adsorbedand stuck to the wall surface, freely moved through the pore space, orexited the membrane, as shown in FIG. 19. Again, only a few major flowconduits appear. Several interesting points include within the 100×130μm² (width×length) membrane segment viewed: (i) only three major fastflowing fluid channels were present containing fewer particles, (ii)many particles were held-up near the top face of the membrane in the lowflowing regions, and (iii) many pores of the membrane had low or no flowand were devoid of particles. These results suggest that there areopportunities to design better membrane structures.

In FIG. 19 (2 μm bins were used), the integrated particle numbers ofboth the large (2 μm diameter) and small (1 μm diameter) particles as afunction of depth along the membrane from inlet to exit are displayed.This embodiment allows one to estimate the selectivity betweensufficiently distinguishable species (particle here) and solute flux asa function of flow path location. Since the mean pore size of themembranes used here was 5 p.m, the particles were either too small (1and 2 82 m) to exhibit differential selectivity, or the sensitivity ofthe method under these circumstances was inadequate.

In some embodiments, fluid and particle drag through a hypotheticalmicroporous membrane was analyzed. In some embodiments, transient dualparticle tracking in a teardrop membrane was performed. The choice ofhypothetical structures is important and will depend on the desirablegoals of capturing or passing suspended particles in or through amembrane and evenness of flow across the horizontal exit axis. In someembodiments, teardrops were selected because they exhibit relatively lowpressure drop and sufficient surface area for binding. FIG. 20A-C showsa binary system of particles in a membrane with three orientations ofteardrops: ±10°, ±70°, and reversed 10° (=)±170° teardrops relative tothe axial direction of flow, respectively. The membrane is a computergenerated, staggered array of teardrop obstructions. In someembodiments, individual rows of teardrops were rotated ±170° from thevertical. 300 1 μm diameter and 150 2 μm diameter particles wereintroduced together and simultaneously in a 1 mM electrolyte solution inthe area directly above the top membrane surface. The membrane wallpotential φ₀=−65 mV, the particle charge was Z=−100, andparticle-particle interactions included van der Waals (Hamaker constant,A=6×10⁻²¹ J). The driving pressure at the top face was 14 kPa (˜2 psi)and the maximum velocity was 0.2 m/s.

In some embodiments, binary particle tracking was performed. Results ofparticle tracking for a binary system flowing through the commercialmembrane and a computer-generated teardrop membrane are shown in FIGS.21A and 21B, respectively. In some embodiments, teardrops were selectedbecause they exhibit relatively low pressure drop, have sufficientsurface area for binding, have more uniform velocity distribution,permit an overall average velocity nearly twice that of the commercialmembrane, and particle tortuosity and hence movement was controlledthrough teardrop orientation and wall charge. In some embodiments, 300 1μm diameter and 150 2 μm diameter particles were simultaneouslyintroduced in the region directly above the top membrane surface. Amembrane wall potential of φ₀=−65 mV and a transmembrane pressure of 14kPa (˜2 psi) with van der Waals attraction to the surface were selected.Their lateral (to the flow axis) positions were randomly selected, andthey were given initial velocities and trajectories dictated by thelocal flow field. The particles were then subjected to drag, electrical,and van der Waals forces. When the particles approached the membrane tothe point where van der Waals forces overcame the electrostatic andhydrodynamic forces, they stuck to the surface.

The velocity profiles in the two membranes are much different. Theteardrop system was designed to have a more uniform velocitydistribution, as shown in FIG. 21. Though the maximum velocity was muchless than in the commercial membrane, the particles, in the teardropmembrane, were more easily caught up in the flow and dragged toward theoutlet of the membrane because the overall average velocity was nearlytwice that of the commercial membrane (0.077 vs 0.04 m/s).

In some embodiments, tortuosity and chemical surface effects wereanalyzed. To investigate the effect of tortuosity, threecomputer-generated teardrop membranes were constructed with orientationsof ±170°, ±10°, and ±45° relative to the axial direction of the flow, asshown in FIGS. 22A-C, respectively. 300 1 82 m diameter and 150 2 82 mdiameter particles were simultaneously introduced in the same solutionas described in FIG. 21B. The orientation of the structures clearlyaffected the motion of the particles through the membrane as shown bythe particle penetration distributions associated with each image. Insome embodiments, the particles moved more cohesively through thestructure when the bulbous end of the teardrop was oriented towards themembrane inlet (FIG. 22B versus FIG. 22A). Providing a more tortuouspath (FIG. 22C) served to spread the particles a bit more and offered ahint at the separation between the large and small particles. Moreparticles were retained (stuck) in this design than the other designs.Thus, the embodiment shown in FIG. 22C shows that an even more tortuouspath would be required for further particle separation.

The results of parametric study on the effect of van der Waals forcesbetween the particles and a computer-generated teardrop membrane areshown in FIG. 23. The computer-generated teardrop membrane was the sameas that shown in FIG. 21B with φ₀=−65 mV and a transmembrane pressure of14 kPa (˜2 psi). 300 1 82 m diameter (small circles) and 150 2 82 mdiameter (large circles) particles were simultaneously introduced in thesame solution as described in FIG. 21B. The Hamaker constant, whichranged from 6×10⁻²¹ to 6×10⁻¹⁷ J, is a measure of the van der Waalsattractive force between the particles and the membrane surface, whichdepends on the particle diameter and the square of the distance betweenthe particle and the wall. In some embodiments, the flow field was heldconstant to provide insight on the relative strength of particle-wallattractive forces on particle rejection. Particle rejection for the twosizes of particles used in this embodiment are shown in FIG. 23A. Therejection was a strong function of the strength of the intermolecularforces though a relatively weak function of particle size. FIG. 23Bshows the average particle velocity for the slowest and fastest movingparticles through the 130 82 m membrane structure. As used herein, theaverage velocity is defined as the membrane thickness divided by theparticle residence time. The fastest particle velocity was a weakfunction of the Hamaker constant. This occurs because for this fractionof particles, hydrodynamic forces govern the motion of the particles andthese particles occupy regions in the center of the channels forming theopen space of the membrane. The slowest particles, where hydrodynamicforces exert much less influence, are greatly affected by the presenceof the membrane solid surface. These particles generally adsorb onto themembrane surface. As the intermolecular forces increase, the averagevelocity of the slower particles also increases. This occurs because theslowest particles adhere to the membrane surface and are removed fromthe average calculation. At large values of the Hamaker constant,representing strong particle-wall interactions, as more slow-movingparticles are captured, the slowest free moving particle residence timeapproached that of the fastest moving particle, as shown in FIG. 23C. Insome embodiments, the velocity of the fastest moving particle had alsoincreased. These results indicate a regime change where intermolecularforces begin to affect all particles within the channels of thesynthetic membrane. The boundary delineated by a plot of residence timeversus Hamaker constant for the slowest moving particles establishes aphase diagram where above the boundary, particles are captured by themembrane surfaces, and below the boundary, particles permeate entirelythrough the membrane, as shown in FIG. 23C.

FIG. 24 shows a comparison between the commercial membrane and thehypothetical teardrop structure membrane embodiments. The membrane is acomputer-generated, staggered array of teardrop obstructions. Individualrows of teardrops were rotated ±170° from the vertical. 300 1 82 mdiameter and 150 2 82 m diameter particles were introduced together andsimultaneously in a 1 mM electrolyte solution in the area directly abovethe top membrane surface. The membrane wall potential φ₀=−65 mV, theparticle charge was Z=−100, and particle-particle interactions includedvan der Waals (Hamaker constant, A=10-15 J). The driving pressure at thetop face was 14 kPa (˜2 psi) and the maximum velocity was 0.2 m/s.

In some embodiments, the computational predictions with the commercial 5p.m mean pore size microporous membrane were compared qualitativelyagainst the mean particle number results obtained from microfiltrationusing the actual 5 82 m mean pore size microporous membrane with a feedcontaining 0.005% SiO₂ particle suspension (of 1 82 m mean diameter), asshown in FIG. 25. Using SEM to image the porous flow region with EDSoverlay of the silicon signal to specifically identify the SiO₂particles, SiO₂ particles residing within the membrane were located andcounted with distance from the top face to the bottom face of themembrane (orthogonal to the main flow direction), as shown in FIGS.25A-C. FIGS. 25D-E show a qualitative comparison of the number of SiO₂particles from the filtration experiments (data is binned every 2 82 mor 40 pixels along flow axis) with the computer-generated profile ofpredicted particle number as a function of membrane depth (x-coordinate)using computational fluid mechanics, respectively. Both sets of resultsshow that the particles were held-up in the first 2-30 μm of the flowpath and are qualitatively similar.

Embodiments of the present technology can improve the performance(including the selectivity and/or capacity) of a synthetic membranethough optimal design of the membrane structure and chemistry, species(e.g., particles) transport inside a porous membrane under appliedpressure is needed. Some embodiments of the technology include acombined 2D computational fluid and particle drag mechanics model withintermolecular force measurements to study particle intrusion andattachment in the pores of a commercial microporous polymer membrane.Besides providing insight into particle capture and fluid flow withinmembrane pores, the 2D model qualitatively agrees with filtrationexperiments, predicts a few dominant flow paths, excessive capture ofparticles near the entrance or top face of the membrane, and largernumbers of particles in the slower flowing regions. This technology isused to assess the performance of membranes, in some embodiments. Someembodiments show that the internal 2D morphological structure ofcommercial microporous membranes are poorly designed for optimal fluidflow and particle capture or passage, and hence selectivity andpermeation flux. An embodiment of a 2D model according to thistechnology improves filtration performance. Additional embodimentsinclude three synthetic morphological structures based on a teardropdesign that predict the average axial particle velocity and show evenexit axial flow across the lateral exit axis and separation betweensmall and large particles, both of which were not observed forcommercial microporous membranes. Embodiments also demonstrated thatwhen tortuosity increased, the particle transport was delayed, and whenwall attraction increased, with higher Hamaker constants, the fractionof particle capture increased.

Although the technology has been described and illustrated with respectto exemplary embodiments thereof, it should be understood by thoseskilled in the art that the foregoing and various other changes,omissions, and additions may be made there and thereto, withoutdeparting from the spirit and scope of the present technology.

1. A method of assessing a membrane, comprising the steps of:calculating fluid dynamic characteristics of at least one of a membraneand a material to be passed through the membrane from an inlet end ofthe membrane to an outlet end of the membrane, where the materialcomprises particles; obtaining characteristic of at least one forceacting on the particles of the material to be passed through themembrane due to the interaction between the particles and the membrane,the at least one force being an intermolecular force; combining thecalculated fluid dynamic characteristic and the obtained characteristicsto assess the flow of the material through the membrane; and optimizingat least one characteristics of the membrane in relation to the materialwherein the membrane comprises a plurality of rows and a plurality ofteardrop structures arranged in the plurality of rows; and wherein theteardrop structures in each row are arranged such that a bulbous end ofeach teardrop structure is oriented toward the outlet end of themembrane.
 2. The method of claim 1, wherein the step of calculatingfluid dynamic characteristics comprises computation of the fluid andparticle drag mechanics associated with the material in at least twospatial dimensions.
 3. The method of claim 1, wherein the step ofcalculating fluid dynamic characteristics comprises computation of thefluid and particle drag mechanics associated with the material in threespatial dimensions.
 4. The method of claim 2, wherein the step ofcalculating fluid dynamic characteristics comprises computation of thefluid and particle drag mechanics associated with the material in threespatial dimensions.
 5. The method of claim 1, wherein the step ofobtaining characteristics of at least one force comprises measuring theintermolecular forces between the membrane and the particles.
 6. Themethod of claim 1, wherein the step of optimizing at least onecharacteristics comprises optimizing the capture or release of particlesby the membrane.
 7. (canceled)
 8. The method of claim 1, wherein theteardrop structures in each row are arranged at substantially the sameangle with respect to an anticipated direction of flow through themembrane.
 9. The method of claim 8, further comprising that rows of theteardrop structures in which the structures are at an angle of 10°alternate with rows of the teardrop structure in which the structuresare at an angle of −10° relative to the anticipated direction of flowthrough the membrane.
 10. The method of claim 8, further comprising thatrows of the teardrop structures in which the structures are at an angleof 45° alternate with rows of the teardrop structure in which thestructures are at an angle of −45° relative to the anticipated directionof flow through the membrane.
 11. The method of claim 8, furthercomprising that rows of the teardrop structures in which the structuresare at an angle of 70° alternate with rows of the teardrop structure inwhich the structures are at an angle of −70° relative to the anticipateddirection of flow through the membrane.
 12. The method of claim 8,further comprising that rows of the teardrop structures in which thestructures are at an angle of 170° alternate with rows of the teardropstructure in which the structures are at an angle of −170° relative tothe anticipated direction of flow through the membrane.
 13. The methodof claim 1, wherein the membrane is formed of a microporous hydrophilicpolymer material.
 14. A microporous membrane, comprising a plurality ofrows and a plurality of teardrop structures arranged in the plurality ofrows, wherein the teardrop structures in each row are arranged atsubstantially the same angle with respect to an anticipated direction offlow through the membrane from an inlet end of the membrane to an outletend of the membrane, and wherein the teardrop structures in each row arearranged such that a bulbous end of each teardrop structure is orientedtoward the outlet end of the membrane.
 15. (canceled)
 16. Themicroporous membrane of claim 14, further comprising that rows of thestructures in which the structures are at an angle of 10° alternate withrows of the structure in which the structures are at an angle of −10°relative to the anticipated direction of flow through the membrane. 17.The microporous membrane of claim 14, further comprising that rows ofthe structures in which the structures are at an angle of 45° alternatewith rows of the structure in which the structures are at an angle of−45° relative to the anticipated direction of flow through the membrane.18. The microporous membrane of claim 14, further comprising that rowsof the structures in which the structures are at an angle of 70°alternate with rows of the structure in which the structures are at anangle of −70° relative to the anticipated direction of flow through themembrane.
 19. The microporous membrane of claim 14, further comprisingthat rows of the structures in which the structures are at an angle of170° alternate with rows of the structure in which the structures are atan angle of −170° relative to the anticipated direction of flow throughthe membrane.
 20. The microporous membrane of claim 14, wherein themembrane is formed of a hydrophilic polymer material.